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I mkm Agriculture 
■ Canada 



cooe 



^GKiOJLTUKc- CANAOA 



23/04/85 



NO. 



Research Direction generale 
Branch de la recherche 



Contribution 1983-14E 



LIBRAHY/BI8LI0THEQUE UT 



T ^A KIM UC5 




Sequential sampling for 
pest control programs 



630.72 
C759 
C 83-14 
c.2 
OOAg 








Canada 






The map on the cover has dots 
representing Agriculture Canada 
research establishments. 



Sequential sampling for 
pest control programs 



GUY BOIVIN 

Research Station 
Saint-Jean-sur-Richelieu, Quebec 

CHARLES VINCENT 

Research Station 
Saint-Jean-sur-Richelieu, Quebec 

Project No. 923, Saint-Jean-sur-Richelieu 



Research Branch 
Agriculture Canada 
1983 



Copies of this publication are available from: 

Dr. G. Boivin 

Research Station 

Research Branch, Agriculture Canada 

Saint- Jean-sur-Richelieu, C.P. 457 

Quebec 

J3B 6Z8 

Produced by Research Program Service 

©Minister of Supply and Services Canada 1983 



ABSTRACT 



In comparison with traditional sampling techniques, sequential sampling reduces 
the number of samples required to evaluate a population level by 40-80%. This 
reduction lowers monitoring costs and facilitates the implementation of an 
integrated pest management program. This paper reviews four prerequisites 
imposed by this technique and compares the advantages and limitations of the 
methods used by Wald in the United States and by Iwao in Japan. It also 
describes the calculation methods used and provides a practical example of 
the use of Poisson and negative binomial distributions. 



RESUME 



La technique de 1' echantillonnage sequentiel permet de reduire de 40 a 80% le 
nombre d 1 echantillons necessaires a 1' evaluation d'un niveau de population par 
rapport aux techniques traditionnelles d' echantillonnage, favorisant ainsi la 
mise en place d'un programme de lutte integree . Les quatre elements prerequis 
a 1' utilisation de cette technique ainsi que les avantages et les limites de 
l'approche americaine de Wald et de l'approche japonaise d'lwao sont presentes. 
Les methodes de calcul et un exemple pratique illustrant 1' utilisation de cette 
technique d' echantillonnage sont donnes pour les distributions de Poisson et 
de la binomiale negative. 



CONTENTS 
INTRODUCTION / 3 

PREREQUISITES / 4 

Sampling techniques / 5 
Economic threshold / 5 
Spatial distribution / 5 
Error levels / 7 



PRINCIPLES OF SEQUENTIAL SAMPLING / 7 

Wald's procedure / 7 

Probability curve / 7 
Average sample number curve / 9 
Calculation of acceptance boundaries / 9 
End of sampling / 14 

Iwao's procedure / 14 

Curve of the upper acceptance limit / 15 
Curva of the lower acceptance limit / 15 
Maximum number of samples / 15 



USING THE SAMPLING PLAN / 17 

Wald's procedure / 17 

Y intercept of D-^ / 17 

Y intercept of D 2 / 17 
Slope of the lines / 20 
Probability curve / 20 
Average sample number curve / 20 

Iwao's procedure / 20 

Upper acceptance limit / 21 
Lower acceptance limit / 21 
Maximum number of samples / 21 



CONCLUSIONS / 21 
GLOSSARY OF SYMBOLS / 23 
REFERENCES / 25 



INTRODUCTION 

Problems related to the use of massive amounts of chemicals in pest control 
have served to create an increasing awareness of the importance of diver- 
sifying existing pest control programs. The theory of integrated management 
can provide a solution, by offering a series of steps that can be taken before 
having to resort to chemical treatments (Luckman and Metcalf 1975) . 

The implementation of an integrated pest management (IPM) program implies a 
reduction in the number of chemical treatments applied and in the amount of 
chemicals used per treatment. Reducing the amount of chemicals applied 
decreases the risk that the pest population will develop a resistance to the 
product and generally leads to a decrease in the cost of production. However, 
to ensure the success of this type of program, pest population levels must be 
monitored throughout the growing season (Boivin and Vincent 1981) . 

Monitoring the levels of the pest populations before or after treatment 
requires a regular program of field sampling, which involves a significant 
commitment in terms of human and economic resources. Nevertheless, sampling 
remains essential to the IPM program, providing information on the crop under 
study. Therefore, the grower or some other qualified person should learn how 
to sample in such a way as to obtain this information as efficiently as 
possible . 

Ruesink and Kogan (1975) describe two sampling techniques for estimating 
population levels. The spatial distribution of the organism can be used to 
determine the optimal number of samples that must be obtained to estimate the 
average density of the pest population within certain predetermined limits of 
error (Karandinos 1976). This number of samples represents a compromise: it 
is too high when the population level is high, and too low at low population 
densities. When the population level is near the economic threshold, there is 
a maximum probability of reaching the wrong conclusions (Fohner 1981) . 

As the purpose of sampling is to make recommendations for intervention, the 
aim is simply to determine whether the pest population is above or below a 
certain economic threshold. Sequential analysis makes it possible to arrive 
at this objective while reducing the number of samples required. 

Sequential sampling was developed during the Second World War (Wald 1943, 
1945, 1947). Its great success in programs used to control the quality of 
military equipment led to its classification as a 'military secret' in the 
United States until 1945, when it was finally made public. Shortly after that, 
it was applied to forest entomology (Morris 1954, Waters 1955) and later to 
agricultural entomology (Sylvester and Cox 1961, Harcourt 1966a, b) . Since 
then, sequential sampling has been used to determine the population levels 
of insects in many different crops (Pieters 1978) . 



Sequential sampling evolves by successive stages and the decision to inter- 
vene may be made after each sampling. Once a critical level is reached, 
namely the economic threshold, sampling stops and a recommendation is made. 

In cases of heavy infestation, several samples may show that the pest 
population still remains above the economic threshold. On the other hand, 
if, after several samples, no pests are captured, one may be reasonably sure 
that the population level is low. By virtue of this principle, sequential 
sampling makes it possible to make quick decisions within a preestablished 
margin of error, reducing the number of samples required by 47-63% (Wald 
1947) , or even up to 79% in some cases (Pieters and Sterling 1974) , in 
comparison with the more traditional sampling techniques described by 
Cochran (1977) . 

This method can be used in all plant protection disciplines that require 
estimations of population levels, particularly in entomology, plant pathology, 
and nematology. 

Sequential sampling can also be used to evaluate the effectiveness of 
insecticide treatments in the field and to determine whether parasite and 
predator populations are high enough to avoid an intervention. Kuno (1969, 
1972, 1977) has used this technique to estimate population means within known 
error limits. Johnson (1977) has estimated sex ratios in insect groups 
captured with sticky traps by examining them sequentially. Dichotomized data 
(e.g. sex ratios, infected versus healthy plants) can also be treated with 
this technique. Thus, sequential sampling has become a very useful tool for 
determining the optimal utilization of the resources required to evaluate 
population parameters. 

The purpose of this paper is to show the advantages and disadvantages of the 
techniques used by Wald (1947) in the United States and by Iwao (1975) in 
Japan. 



PREREQUISITES 

Four elements are necessary to establish a sequential sampling program: 

. a practical and reliable sampling procedure 

. the economic threshold of the organism in -a particular crop 

. the parameters of a mathematical model describing the spatial distri- 
bution of the sampled organism 

. realistic and acceptable error levels for estimating pest populations 
in crops. 

The following is a review of these prerequisites. 



SAMPLING TECHNIQUES 

Sampling techniques can be either direct (when the pest is captured) or 
indirect (when using effects left by the organism, such as waste products 
or indications of damage) . The same technique must be used to obtain the 
economic threshold and the spatial distribution of the organism. When more 
than one developmental stage is studied, the relative effectiveness of the 
technique used must be determined so that the estimates can be adjusted by 
means of appropriate weight factors. 

ECONOMIC THRESHOLD 

The economic threshold is the population level above which an intervention 
becomes necessary (Stern 1973) . 

For better precision, this threshold should be established for each cultivar, 
because the susceptibility of plants varies from one cultivar to another. 
Thus, the economic threshold is a function of crop value and pest control 
cost . 

The use of Wald's method (1947) requires the establishment of two critical 
population levels: Ai (level below which no treatment is required) and 
X2 (level above which treatment is recommended) 1 . The choice of the interval 
between A]_ and A 2 is based on the biology and behavior of the pest, crop value, 
and the potential damage that may be caused by the pest (Waters 1974) . The 
probability of arriving at a correct classification is lower when the popu- 
lation level is between A-, and A2 than when the population level is below 
A-^ or over A 2 (Fohner 1981) . The smaller the interval between A, and A2, 
the larger the number of samples required. 

When the economic threshold is unknown, a sequential sampling plan can be 
implemented on the basis of a preliminary threshold, called the action level 
by Lincoln (1978) . This provisional level may be used with certain reserva- 
tions regarding the precision of the sampling or the recommendations made. 

SPATIAL DISTRIBUTION 

The spatial distribution of the pest population determines the number of 
samples required to arrive at a predefined level of precision. The calculation 
of the limits of this zone of acceptability varies according to whether the 
distribution of the organism is uniform (Fig. 1A) , random (Fig. IB), or 
contagious (Fig. 1C) . 



Mathematical symbols are defined in the glossary at the end of 
this bulletin. 



s 2 <x 



B 






• • 
• 

• 


• 


• • . . • 

• 
• • • • 


• 


• 


• • • • 


• 
• 


• 


• • # • 



s 2 =x 



• • • • • 

••• • • .. • 

• • • • 

• • • • • • 



2 - 
S > X 



gb. 



Fig. 1. (A) Uniform, (B) random, and (C) contagious distributions. 



Two methods can be used to characterize the spatial distribution of an 
organism. The first consists of obtaining samples and then comparing 
the frequency distribution of the captured organisms with theoretical 
distributions such as Poisson, Poisson binomial, or negative binomial. 
The fit between the observed and theoretical frequency distributions can 
be quantified by tests such as G, X » or Kolmogorov-Smirnov (Sokal and 
Rohlf 1981) . 

The second method characterizes the spatial distribution of an organism by 
using the mean crowding index (Lloyd 1967) in relation with Iwao's regression 
technique (1968) . This method makes it possible to describe spatial distri- 
bution on the basis of two parameters (Iwao 1977) and implies a calculation 
method that is independent of the number of samples obtained. 



ERROR LEVELS 

In any sequential sampling plan, two statistical errors can be made. First, 
the population level may be determined to be below a certain critical threshold 
while in reality it is above; consequently one would fail to recommend treat- 
ment when it is necessary. This error is known as a Type I error and it has 
a probability of a. Second, the population may be overestimated, resulting 
in recommendations for unnecessary treatment. This constitutes a Type II 
error, where the probability is 3. 

It is more serious to fail to recommend a necessary treatment (Type I) than 
to recommend an unnecessary action (Type II) , because the cost of the treatment 
is lower than the potential losses. Consequently, the probability of committing 
a Type I error (a) should be lower than the probability of committing a Type II 
error (3) . The precision of the sampling plan increases as probabilities of 
errors I and II decrease, but the number of samplings required may become 
prohibitive. Most sequential sampling programs have error levels of about 
0.05 to 0.1 (Sevacherian and Stern 1972; Pieters and Sterling 1974, 1975; 
Gruner 1975 J Strayer et at. 1977) , and sometimes of 0.4 (Danielson and Berry 
1978, Burts and Brunner 1981) . When error level a is 0.05, there will be, 
on the average, one Type I error in every 20 decisions. 



PRINCIPLES OF SEQUENTIAL SAMPLING 

WALD'S PROCEDURE 

Wald's sampling method (1947) is based on a theoretical mathematical distri- 
bution that can describe the observed spatial distribution of the insect 
population under consideration. 

Probability curve 

For each level of infestation, the probability curve, also named operating 
characteristic curve (Onsager 1976), indicates the probability of accepting 
hypothesis Hj , which assumes that the mean of the population sampled is 
equal to or below the value of X-j_ (Oakland 1950) (Fig. 2) . The probability 



I 
,ss 

'35 
O 

a 

o> 

c 

& 

o 
o 
re 

•4- 

o 



JQ 

re 

JQ 
O 




0.2- 



Sample mean 



Fig. 2. Probability curve of a sequential sampling plan, using Wald's method 



of accepting hypothesis H^^that is, that the population mean is higher than A2 , 
follows an inverse curve. The slope of this curve depends on the values 
chosen for a and 3. 

Regardless of what the spatial distribution of the pest is, four points are 
calculated for interpolating the curve. These points are obtained as follows: 



(1) for A =0 


LP = 1 




a' = X 


LP = 1 - a 




x' = b 


LP = a 2 / a 2 - 


■ a l 


A = A 2 


LP = 3 





where LP = level of probability 
ai = Y intercept of D, 
ay = Y intercept of D2 
b = slope 



Average sample number curve 

This curve makes^it possible to predict the mean number of samples that must 
be obtained before making a decision (Fig. 3). The number of samples varies 
with the level of infestation and is at a maximum near the economic threshold. 
If the prediction regarding the number of samples required exceeds available 
resources in terms of time and labor, the only remaining alternative is to 
sacrifice precision by increasing either the probability of errors I and II 
or the difference between A., and X „ . 

The equations used to calculate four of the points in this curve are described 
below. The ordinate at the origin of this curve indicates the minimum number 
of samples that must be obtained before making a decision (Fig. 3). 

Calculation of acceptance boundaries 

Sequential sampling equations are given for two types of spatial distributions 
frequently found in pest control situations: Poisson and negative binomial 
distributions. The equations are from Wald (1947), Waters (1955), and Onsager 
(1976), who also give other equations that can be used with other types of 
distribution. 

Poisson Distribution - This model is a probability distribution whose mean and 
variance have a common constant value. It is used to describe a population 
that is randomly distributed in space (Fig. IB) (Southwood_1978) . The only 
measurable parameter required to describe it is the mean (X) . 



z 

< 



0) 

E 

3 
C 

a 
E 

<0 
«/> 

V 
O) 
<0 

k- 

> 

< 




Sample mean 



Fig. 3. Average sample number curve of a sequential sampling plan obtained 
by Wald's method. 



10 



Y intercept of D. (Fig. 4): 



1 

1 - a 
log ( 5 ) 



(5) a ± = - 



X 2 



Y intercept of D„ : 

1-3 
log (— — > 



(6) a 2 = 



log ( X? ) 

X_ 



Slope of the two parallel lines 
0.4343 (X 2 - X x ) 



(7) b = 



X-» 
log (-*-) 

X, 



where a = probability of making a Type I error 

3 = probability of making a Type II error 

X-j_ = population mean used as lower limit 

X 2 = population mean used as upper limit . 



Average sample number curve (ASN) 

(8) ASN = LP (a x - a2> + a 2 

X' - b 



where LP = chosen level of probability at X-^ 
X' = population mean 



There are two particular cases where the calculations are made in a different 
way: 

where X' = 0, LP = 1, and ASN = a^-b 

X* = b, ASN = a 1 a 2 /-b 



11 



The peak of this curve indicates the maximum number of samples to be 
obtained, on the average, when the population density is near the economic 
threshold. This point is useful in deciding when to stop sampling in cases 
where no decision has been made (see "End of Sampling") . This peak is 
near the three values of ASN calculated for A' = 0, A 1 = At , and A' = b. 

Negative Binomial Distribution - This model is used to describe a contagious 
distribution (Fig. 1C) (Southwood 1978). This distribution frequently 
observed in pest populations can be described by two parameters: the 
arithmetic mean and K, a constant used to measure the degree of aggregation 
of the population. 

Acceptance boundaries based on a negative binomial distribution are calculated 
on the basis of four new parameters, as follows: 

(9) P 1 = A 1 /K 

(10) P 2 = * 2 / K 

(11) Q 1 = 1 + P 1 

(12) Q 2 = 1 + P 2 



That K is constant, regardless of the level of infestation, is one of the 
underlying hypotheses in the use of sequential statistics. If the value of 
K increases with the sample mean, a common K, denoted K c , can be calculated 
for all the means (Bliss and Owen 1958) . If K c has not been determined for 
the pest population under consideration, it is necessary to ensure that K is 
almost constant for the range of means covered by the sequential sampling 
plan (Onsager 1976) . 

Y intercept of D-^ (Fig. 4): 

1 - a 



(13) a ± = - 



log ( — — ) 



P 2<*1 

log ( ) 



P 1 Q 2 



Y intercept of D~ 



l - 6 
lQ g (— n — ) 

(14) a_ = 

log (-^-) 

p lQ2 

Slope of the two parallel lines: 

log <-9*> 

(15) b = K — 



P 2 Q 1 
108 ^ 



12 



28i 




Number of samples taken (tapped branches) 



Fig. 4. Acceptance boundaries of a sequential sampling plan according to Wald's method. 



13 



Average sample number curve (ASN) : 

a ? + (a.. - a„) LP 
(16) ASN = — — 



X' - b 

There are two special cases where the calculations are obtained in a different 
way: 

where A' = 0, LP = 1, and ASN = a /-b 

A' = b, ASN ■ a a 2 /-(b 2 /K + b) 

When a is equal to or smaller than 3, the peak of the ASN curve approaches 
the value of ASN calculated for A' = b. As the value of a increases over 
the value of 3, the accuracy of the estimate decreases, and the exact value 
must be found by iteration by means of equation 16 (Onsager 1976) . 

End of sampling 

When the actual mean of the population falls between the two chosen limit 
values (Ai and A^) , it is possible to take a large number of samples without 
going outside these limits and remain incapable of reaching a decision. 
Therefore, a mechanism must be provided to make it possible to end the 
sampling process. 

Wald (1947) proposed a mathematical solution that takes into account changes 
at the level of a and 3 errors, but this solution requires complex mathematical 
calculations. Waters (1974) suggested that sampling should stop when the 
maximum number of samples predicted by the average sample number curve has been 
reached. Nevertheless, he does not explain how we can choose between hypotheses 
H-l and H2 Once the sampling has stopped. 

Some authors suggested that sampling should be started again at a later point in 
time (Sevacherian and Stern 1972) or that the hypothesis represented by the 
acceptance boundary closest to the last sampled point should be accepted 
(Sterling and Pieters 1974, 1975). 



IWAO'S PROCEDURE 

Mean crowding (Lloyd 1967) is an aggregation index that can be obtained as 
follows: 

* 2 
(17) X « X + (— - 1) 
X 

* 

The mathematical relationship between mean density (X) and mean crowding (X) 



14 



describes certain characteristics of the spatial distribution that are 
inherent to each species in a given habitat. Iwao (1968) demonstrated that 
this relationship can be described by a simple linear regression. 

There are two parameters that describe the type of spatial distribution of 
the organism: the Y intercept of the regression line (a r ) , that is, the 
index of basic contagion; and the slope of the regression (b r ) , that is, 
the density-contagiousness coefficient. The first of these parameters 
characterizes the basic unit of the population, whereas the second describes 
the distribution of these units in space. Regression validity must be checked 
in terms of the significance of the correlation coefficient (Steel and Torrie 
1980) . 

Contrary to Wald's method, the economic threshold is used directly to calculate 
the limits of the acceptance curves. The following equations are from Iwao 
(1975) and Southwood (1978) . 

Curve of the upper acceptance limit 



(18) C = N x ET + t * N [(a + 1) ET + (b - 1) ET 2 ] 



Curve of the lower acceptance limit 



. 

(19) C. = N x ET - t N [(a + 1) ET + (b - 1) ET Z ] 

where C = total captures 

N = number of samples taken 

ET = economic threshold 

t = value of Student's t at chosen level of significance for a two-sided 

test and an infinite number of degrees of freedom 
a = index of basic contagion (Y intercept) 
b = density-contagiousness coefficient (slope) 

Two curves are obtained by calculating several C s and C^ for different values 
of N (Fig. 5) . The space between these two curves increases with the amplitude 
of the degree of precision. If the population mean of the target population 
is equal to the economic threshold, a large number of samples can be obtained 
in between the calculated limits. Iwao's procedure makes it possible to cal- 
culate the maximum number of samples that must be taken in order to determine 
if the population level is equal to the economic threshold, within a predeter- 
mined confidence band. 



Maximum number of samples 

t 2 2 

(20) N = -E- [(a + 1) ET + (b - 1) ET ] 
(.max) jz r r 

where d = confidence interval of the estimated mean density (see example) 



15 




Number of samples taken (branches tapped) 



Fig. 5. Acceptance curves of a sequential sampling plan, according to Iwao's method. 



16 



This procedure takes into account the possibility that the mean and economic 
threshold might be the same. When sample N_„__ is reached, one decides that 
the population mean is at the economic threshold and a decision can then 
be made. 

USING THE SAMPLING PLAN - EXAMPLE 

We have so far described the sampling plan in graphic terms. Some authors 
(Onsager 1976, Mason 1978) have examined the difficulties involved in using 
graphic methods in the field and have proposed the use of tables (Tables 1 
and 2) . For each sampling intensity, the table gives the values of cumulated 
captures for each limit. The cumulative captures are compared to the lower 
and upper limits at the appropriate number of samples. 

An example is presented here wherein the methods of Wald and Iwao can be 
practically compared. This example uses the sampling technique and spatial 
distribution used with early nymphs of Lygocoris communis (Knight) (Hemiptera: 
Miridae) (Boivin 1981) . This sampling technique, whose effectiveness and 
reliability have already been evaluated, consists of tapping apple branches 
over a white cloth measuring 1 m^ . A provisional economic threshold of one 
first- or second-instar nymph per branch tapped is used. When the population 
density is above this threshold, treatment is required, while populations 
below this level are tolerated. 

WALD'S PROCEDURE 

The spatial distribution of early Lygocoris communis nymphs is described by a 
negative binomial distribution with K = 2.13. Population means specified 
as the lower and upper limits are A-j_ = 0.5 and A 2 = 1.5 nymphs per branch; 
error limits are a = 0.1 and f> = 0.2. 

Pi = 0.5/2.13 = 0.2347 
P 2 = 1.5/2.13 = 0.7042 
Q 1 = 1 + 0.2347 = 1.2347 
Q 2 = 1 + 0.7042 = 1.7042 

Y intercept of D 

1 ( °- 9 ^ 

0-2 ! / r 0.653 2 . „ 7n 

a i = " = " !° S ' 7 ,F = " 0.3372 = - 1 - 9370 

1 , 0.7042 x 1.2347 . log 2 ' 1738 

g ^0.2347 x 1.7042 ; 

Y intercept of D_ 

"" \ ,0.8 ; 

log ( 



0.1 ' log 8 m 0.9031 

a 2 ' 0.7042 x 1.2347 log 2.1738 0.3372 ' /,b/ 

8 C 0.2347 x 1.7042 ; 



1/ 



17 



Table 1. Acceptance limits of hypotheses H. and H„ for a sequential sampling 
plan according to Wald's procedure 



Number of samples 
(branches tapped) 



Lower limit 



Upper limit 






1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

25 

30 

35 



STOP SAMPLING 
AND TOLERATE 
THIS POPULATION 
LEVEL 



1 

2 

2 

3 

4 

5 

6 

7 

8 

9 

10 

10 

11 

12 

13 

14 

15 

16 

20 

25 

29 



CONTINUE 
SAMPLING 



4 

4 

5 

6 

7 

8 

9 

10 

11 

12 

12 

13 

14 

15 

16 

17 

18 

19 

19 

20 

25 

29 

34 



STOP SAMPLING 
AND APPLY 
TREATMENT 



18 



Table 2. Acceptance limits of hypotheses Hi and H2 for a sequential sampling 
plan according to Iwao's procedure 





Number of s 


iamples 




Lower limit 


Upper limit 


(branches tapped) 












1 










4 




2 






- 




6 




3 






- 




8 




4 






- 




10 




5 


STOP 


SAMPLING 


- 


CONTINUE 


11 


STOP SAMPLING 


6 


AND TOLERATE 


- 


SAMPLING 


13 


AND APPLY 


7 


THIS 


POPULATION 


- 




15 


TREATMENT 


8 


LEVEI 


4 


- 




16 




9 











18 




10 











19 




11 






1 




21 




12 






2 




22 




13 






2 




23 




14 






3 




25 




15 






4 




26 




20 






7 




33 




25 






10 




40 




30 






14 




46 




35 






18 




52 




40 






22 




58 




45 






25 




64 




50 






29 




70 





19 



Slope of the lines 



b = 2.13 x 



log 1.7042 
1.2347 



0.14 



log ( 0.7042 x 1.2347 ) 
(0.2347 x 1.7042) 



= 2.13 x 



0.3372 



= 0.8841 



These acceptance limits are shown in Fig. 4 and Table 1. 

Probability curve 

Four points on this curve make it possible to interpolate the general 
curve . 

For A' = LP = 1 

A» = 0.5 LP = 0.9 

A' = 0.8841 LP = 0.5803 

A' =1.5 LP =0.2 

This curve is illustrated in Fig. 2. 

Average sample number curve 

Four points are calculated to interpolate a complete curve: 

For A' =0 ASN = 2.1909 

A' = 0.5 ASN = 3.8414 

A* = 0.8841 ASN = 20.0045 

A 1 = 1.5 ASN = 2.8485 

Since a is smaller than 3 (0.1 < 0.2), the maximum value of the mean number 
of samples will be close to the value of ASN for A' = 0.8841, that is, 20 
samples. The minimum number of samples to be obtained before making a decision 
is two samples (Fig. 3). 

This sequential sampling plan makes it possible to decide whether the pest 
population level is above or below the acceptance limits, after obtaining a 
maximum number of 20 samples, on the average. 

IWAO'S PROCEDURE 



The parameters of the spatial distribution of early Lygoaovis communis nymphs 
have been calculated in terms of the regression of mean crowding over the mean. 
In this case, a r = 1.68 and b r = 1.47, with r = 0.91 (significant, a = 0.05). 
The economic threshold is one individual per branch tapped and the error level 
a is 0.1. The values of Student's t, for infinite degrees of freedom, are 
1.64 for a = 0.1 and 1.96 for a = 0.05. 



20 



Upper acceptance limit 

C = N x 1 + 1.64 V N [(1.68 + 1) x 1 + (1.47 - 1) x l 2 ] 
s 

= N + 1.64 V N x 3.15 

Lover acceptance limit 

C ± = N X 1 - 1.64 V N [(1.68 + 1) x 1 + (1.47 - 1) x 1* ] 

- N - 1.64 V N x 3.15 

Acceptance limits of these two curves are shown in Fig. 5 and Table 2. 

It is possible to calculate the maximum number of samples to be obtained 
before estimating whether the population mean is equal to the economic threshold 
An a = 0.1 and d = 0.5 were chosen, which means that at N , the population 
mean is 1 i'0.5 nymph per branch, with an error ot of 0.1. This idea can 
also be expressed as follows: after N samples are obtained, nine times 
out of ten the estimated mean falls within the 0.5 - 1.5 interval. 

Maximum number of samples 

N max = frffi [(1 ' 68 + 1} X X + (1 - 47 " 1} X ]2] 



2.6896 
0.25 



x 3.15 



- 10.7584 x 3.15 

- 33.8890 



Sampling stops after 34 samples have been obtained. We know that, the popu- 
lation level is within the chosen interval and a decision as to whether to 
intervene or not can now be made. 



CONCLUSIONS 

The sequential sampling method makes it possible to determine a pest 
population level with a significant reduction in the number of samples 
required. Available information on the bioecology of the pest determines 
in part the choice of the most appropriate procedure. In our opinion, Iwao s 
procedure has three advantages. 

. It does not require a theoretical mathematical model approaching the 
spatial distribution of the insect. 



21 



. The population mean is evaluated in relation to an economic threshold 
and not an arbitrary interval. 

. Sampling stops when the mean of the population sampled and the error 
levels are known. 



Whatever the procedure chosen, available resources can be used more 
efficiently. Given that the time spent in sampling is one of the main 
problems associated with a detection program, the use of a sequential 
sampling method represents a more attractive systematic detection alterna- 
tive. Significant reduction in costs is another important argument that 
can be used to persuade farmers to follow integrated pest control programs. 
We are thus convinced that sequential sampling is a step forward in the 
implementation of integrated pest management programs. 



22 



GLOSSARY OF SYMBOLS 

a 1 = Y intercept of D (Wald) 

a_ = Y intercept of D„ (Wald) 

ASN = average sample number (Wald) 

a = index of basic contagion, Y intercept of the regression line (Iwao) 
r 

b = slope 

b = density-contagiousness coefficient, slope of regression line (Iwao) 

C. = carve of lower acceptance limit (Iwao) 

C = curve of upper acceptance limit (Iwao) 
s 

d = confidence interval of the estimated mean density (Iwao) 

D 1 = upper acceptance limit of H 1 (Wald) 

D„ = lower acceptance limit of H„ (Wald) 

ET = economic threshold 

H = hypothesis according to which one sample is equal to or below a 
preestablished level 

H_ = hypothesis according to which one sample is equal to or above a 
preestablished level 

K = constant, measure of aggregation 

K = common K 
c 

LP = level of probability (Wald) 

N = number of samples taken 

N = maximum number of samples to be obtained before being able to decide 
max 

whether the mean is equal to the economic level (Iwao) 

P = calculated parameter (Wald) 

Q = calculated parameter (Wald) 

s = sample variance 

t = value of Student's t at chosen level of significance 



23 



X = sample mean 

ft 

X = mean crowding (Iwao) 

a = probability of a Type I error 

(3 =* probability of a Type II error 

A... = mean of sample chosen as lower limit (Wald) 

A = mean of sample chosen as upper limit (Wald) 



24 



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